Anisotropic particle spectra and momentum sharing in multicomponent collision cascades

Nuclear Instruments and Methods in Physics Research B 152 (1999) 252±266

Anisotropic particle spectra and momentum sharing in multicomponent collision cascades Zhu Lin Zhang

1

Huainan Institute of Technology, Department of Electric Engineering 2, Huainan, Anhui 232001, People's Republic of China Received 26 January 1998; received in revised form 3 November 1998

Abstract SigmundÕs analytical sputtering theory has been used to explain various Monte Carlo simulations given by Urbassek and Conrad. It has been demonstrated that only SigmundÕs theory can predict all the simulation results. The present author has shown that all the transport equations describing the statistics of linear random collision cascades are suitable for a complete use of the exact scattering cross-section of power potential …V / rÿ1=m † interaction collisions. Asymptotic expansions of an anisotropic particle ¯ux (L ˆ 0,1) are derived here for an arbitrary multicomponent medium. Several new explicit expressions are further reported for various statistical distribution functions. Asymptotic formulae for energy and momentum sharing are presented. The correction terms in asymptotic expansions of isotropic solutions (L ˆ 0) are evaluated for a binary target by the use of both scattering cross-sections ‰drŠp ˆ C0 dT =T and ‰drŠz ˆ rdT =Tm . It has been shown that ‰drŠz greatly enlarges the range of validity of the asymptotic solutions as compared to ‰drŠp . In some cases, ‰drŠz can even yield exact solutions. As usual, the electronic stopping is ignored in the analysis. Ó 1999 Published by Elsevier Science B.V. All rights reserved.

1. Introduction Since Peter Sigmund ®rst presented the modern sputtering theory in 1969 [1,2], a lot of analytical works on sputtering and related phenomena have been published. These works mainly focused on two aspects. The ®rst aspect is treated papers dealing with a monatomic target. In this case, a number of atomic distribution functions, such as the slowing-down, recoil, scattering and collision 1

Tel. +86 554 6640550. The department subsidized a part of the typing fee and postage for this work. 2

densities, etc., are introduced and derived for a linear atomic collision cascade [3]. Furthermore, taking the anisotropic (or momentum) term (L ˆ 1) into account, these functions have been derived asymptotically [4±6] in order to explain the energy and angular distributions of sputtered atoms [5,6]. It is well known that, for elastic collisions, the isotropic term (L ˆ 0) is asymptotically the anisotropic term proportional to E/E0 , while p (L ˆ 1) is proportional to E=E0 [6]. The second aspect is dealt with in papers treating multicomponent targets [6±8]. In the latter case almost only isotropic atomic distribution functions (L ˆ 0) have been derived for linear atomic collision

0168-583X/99/$ ± see front matter Ó 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 9 8 ) 0 0 9 2 4 - 0

Z.L. Zhang / Nucl. Instr. and Meth. in Phys. Res. B 152 (1999) 252±266

cascades. Special attention was paid to the preferential sputtering from a binary target [7,8]. However, it is to be noted that all of the analytical formulae for both aspects are derived based on rather special interaction cross-sections [1,2,7], and have never been supported either by computer simulation or experimental measurements. In 1992, Urbassek and Conrad developed a new analytical theory [9] (UC theory) to study the energy distributions of recoils in a binary medium cascade by the use of a general form of power cross-sections. In UC theory, the power m is allowed to depend on the collision partners i.e. m ˆ mik , and a wider variety of solutions is found [10]. The results are checked by Monte Carlo computer simulations [9]. Furthermore, in 1993, Urbassek and Conrad, further proposed a general method [11] (VCU theory) to study the energy partitioning and particle spectra in multicomponent collision cascades. By the use of this method, the pertinent integral transport equations are reduced to a computationally much simpler system of di€erential equations. This is even possible for arbitrary particle interaction potentials. The accuracy of this transformation is demonstrated by a comparison with the Monte Carlo simulations. Even though it is straightforward to generalize the analytical theory proposed by Sanders et al. in Refs. [4,5] from a monatomic target to a multicomponent one, almost no such work had been published until 1993 [10]. The intrinsic reason is that, for a binary target, the second term in the asymptotic p expansion (L ˆ 0) varies approximately as E=E0 , which is comparable to the momentum term (L ˆ 1) in magnitude. Therefore, Andersen and Sigmund pointed out that ``the occurrence of poles in the interval between zero and one narrows the range of validity of the asymptotic solution as compared to the monatomic case'' [7]. Recently, taking both (L ˆ 0,1) terms into account, Sigmund et al. derived the analytical formulae for particle ¯ux for a binary target [10,12] with nearly equal masses (M1 M2 ). Thus, these analytical formulae ``are valid only up to ®rst order in the mass difference'' [10]. In the present paper, I wish to develop SigmundÕs theory and widen its applicability, with special emphasis on anisotropic particle spectra

253

and momentum sharing [10,12] in multicomponent collision cascades. In Section 2, I will use SigmundÕs power cross-section [1,2] to approximate the generalized one [9] by choosing a proper power mi for mij and solve the transport equations for particle ¯uxes and recoil densities. It will be shown that SigmundÕs analytical theory can cover all the Monte Carlo simulations in Ref. [9], while, neither UC nor VCU theories could. In Section 3 asymptotic expansions of various statistical distribution functions for L ˆ 0,1, which appeared in the transport theory of sputtering, will be derived by a nonperturbation theory [10,12]. Analytical asymptotic expansions for energy and momentum sharing will be given. In addition, I will propose a new method (SC theory) to build up approximate asymptotic solutions for arbitrary cross-sections. Using the SC theory, some formulae in Refs. [11,12] can be rederived. In Section 4, it will be shown that all the transport equations are suitable for a complete use of both simple power and exact scattering cross-sections of power potential interaction collisions [13]. Therefore, it is natural to gear toward cross-sections intermediate between power-potential scattering [13] and hardsphere scattering [14±16]. The pioneering work of Sigmund et al. [10,12] will be generalized to any multicomponent medium. Special attention will be paid to the second term in the asymptotic expansion of the isotropic solution (L ˆ 0) in order to determine the range of validity of the asymptotic solutions (L ˆ 0,1) in both the Sigmund theory and the theory utilized the exact scattering cross-section (called Present theory in this work and previous one [13]) for a binary target. It will be shown that the Present theory generates a more accurate asymptotic solution than the one given by the Sigmund theory for the same system. In some cases, the Present theory can even yield exact solutions. 2. Sigmund theory Consider a random, Nj ˆ aj N atoms of type atomic mass Mj ) P aj …0 6 aj 6 1; aj ˆ 1† is

in®nite medium with j (atomic number Zj , per unit volume. the concentration of

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Z.L. Zhang / Nucl. Instr. and Meth. in Phys. Res. B 152 (1999) 252±266

j-atoms, and N the atomic density (atoms/cm3 ). t0 …Fij …~ t0 † be the average t;~ t0 †d3~ t;~ t0 †d3~ Let Gij …~ number of j-atoms moving (recoiling) with veloct0 † in a collision cascade ities in the interval …~ t0 ; d3~ initiated by an i-atom starting with an initial velocity ~ t. For not too dense cascades. Gij and Fij obey the linear Boltzmann equations [8,10] h i X Z ak drik Gij …~ t;~ t0 † ÿ Gij …~ t0 ;~ t0 † ÿ Gkj …~ t00 ;~ t0 † k

ˆ X k

1 dij d…~ t ÿ~ t0 †; N t0 Z

ak

i ÿFkj …~ t00 ;~ t0 † ÿ dkj d…~ t00 ÿ~ t0 † ˆ 0;

…2†

t;~ t0 ;~ t00 † is the di€erential crosswhere drik ˆ drik …~ section for scattering between a moving i-atom t00 represent the and a k-atom at rest. Here, ~ t0 and ~ velocities of the scattered and recoiling atom, respectively. The standard procedure for solving Eqs. (1) and (2) was given by Sigmund and Sckerl in Ref. [10]. By going over to energy instead of velocity variables one may write, 1 1 X t;~ t0 † ˆ …2L ‡ 1†HijL …E; E0 †PL …g0 †; …3† Hij …~ 4p Lˆ0 where, Hij represents Gij or Fij ; ~ n ˆ~ t=t, ~ n0 ˆ ~ t0 =t0 , g0 ˆ ~ n
~ n0 and PL …g0 † are Legendre Polynomials. For scattering cross-sections such as drij …E; T † ˆ Qi …E†Aij …X † dX 1 P X ˆ T =Tm > h;

…4†

otherwise

[13], the Laplace transform H~ijL …s† ˆ exp…ÿsu†Hij …E0 eu ; E0 † is found to be ~L …s† ˆ G kj

DLij …s† ˆ ÿbLij …s† for i 6ˆ j X L   bik …s† eLik …s† ÿ dik DLii …s† ˆ

for i 6ˆ j

k

bLij …s†

ˆ aj c

s

Z

1 0

dX Aij …X †X s PL

p X ;

c  cij ; …6a†

…1†

h t;~ t0 † ÿ Fij …~ t0 ;~ t0 † drik Fij …~

drij …E; T † ˆ 0

where DLij …s† is the element (i, j) of the determinant DL …s† and ‰DL …s†Šij is the algebraic cofactor for element (i, j).

L

‰D …s†Šji 1 ; N t0 E0 Qj …E0 † DL …s†

1 1 X L F~ijL …s† ˆ
L ‰D …s†Ški bLkj …s†; E0 D …s† k

R1 0

du
…5† …6†

 h iÿ1 Z 1 eLij …s† ˆ aj bLij …s†
dX Aij …X † 1  0  p Vij ÿ…1 ÿ cX †s PL Uij 1 ÿ cX ‡ p 1 ÿ cX cij ˆ

4Mi Mj …Mi ‡ Mj †

Uij ˆ

2

…Mi ‡ Mj † 2Mi

; and

Vij ˆ

…Mi ÿ Mj † : 2Mi

A simple calculation shows eLij …1 ÿ 0:5L† ˆ …Mj =Mi †0:5L

for L ˆ 0; 1:

…7†

The next step is to evaluate the inverse Laplace transform Lÿ1 ‰H~ijL …s†Š. The main problem is to ®nd the poles and corresponding residues of H~ijL …s† [7]. 2.1. Comparison with Monte Carlo simulations [9,11] In 1992, Urbassek and Conrad presented a wider variety of solutions of linear transport equations in binary media for general power law cross-sections (UC theory), drij …E; T † ˆ Cij Eÿmij T ÿ1ÿmij dT :

…8†

On the other hand, these equations were also solved numerically by means of a Monte Carlo simulation with cij ˆ 1. The authors of Ref. [9] demonstrated that UC theory agreed with the simulation results in the asymptotic regime of small recoil energies. In this work, the present author uses conventional power cross-sections

Z.L. Zhang / Nucl. Instr. and Meth. in Phys. Res. B 152 (1999) 252±266

drij …E; T † ˆ Cij Eÿmi
T ÿ1ÿmi dT ; cij ˆ 1

…9†

to approximate Eq. (8) by choosing a proper mi . The energy E0 is measured relative to the bombarding energy E, i.e. x  E0 =E, and distribution functions are normalized as wij …x† ˆ NC11 E1ÿ2m1
t0 Gij …E; E0 †; fij …x† ˆ EFij …E; E0 †

255

ÿ1 ÿ1 ~ ~ ~ K ˆ ‰~ c21 …C~11 ‡ C~12 †Cÿ1 m1 ‡ C12 …C21 ‡ C22 †Cm2 Š ;

Cm ˆ m=…w…1† ÿ w…1 ÿ m††: The subsequent terms stem from the other poles. It may be sucient to analyse the possibility of poles occurring between zero and one for several special cases simulated by Urbassek and Conrad [9].

to make them dimensionless. From here on, the L in formulae can be omitted as long as L ˆ 0. The values of the coecients in cross-sections (8) and (9) are given in a normalized way as

2.1.1. Detailed balance Parameters in Eq. (8) for Monte Carlo simulations in Ref. [9] are

aj Cij
E2…m1 ÿmj † : C~ij ˆ a1 C11

m22 ˆ 0:20; C~21 ˆ 100; C~22 ˆ 1:

With the help of the above considerations, the inverse Laplace transform of Eqs. (5) and (6) yields  …s ÿ mj † ~ ‰…Cj1 ‡ C~j2 †ejj…s†dij wij …x† ˆ 2x2mj ÿ1 Lÿ1 d…s†  i‡j …10† ÿ…ÿ1† C~ji Š ;

Since C~12 ˆ C~21  C~11 ˆ C~22 , the most frequent collision in the cascade takes place between atom-1 and atom-2. The energy distributions of the two species are closely coupled together and a situation of ``detailed balance'' is established [9]. Therefore, it is natural to choose the following parameters

 ÿ1 ÿ1

fij …x† ˆ x L

1 ~ ~ ‰Cij …Ci1 ‡ C~i2 †eii …s† d…s† 

ÿd…0†dij Š

…11†

 ˆ 1. Furfor L ˆ 0, where i, j ˆ 1, 2,  1 ˆ 2, and 2 ther   s C…1 ÿ mi † C…s† C…s ÿ mi † ÿ ; eii …s† ˆ 1 ‡ mi C…s ÿ mi † C…1† C…1 ÿ mi † h  i d…s† ˆ C~11 ‡ C~12 e11 …s† ÿ C~11 h  i
C~21 ‡C~22 e22 …s† ÿ C~22 ÿ C~12 C~21 :

…12†

Obviously, the leading terms of Eqs. (10) and (11) stems from the highest pole s ˆ 1, wij …x† ˆ 2K…1 ÿ mj †C~jj
x2mj ÿ2  wj …x†;

…13†

fij …x† ˆ KC~jj
…C~j1 ‡ C~j2 †xÿ2  fj …x†;

…14†

where

m11 ˆ 0:30;

m12 ˆ 0:40; m21 ˆ 0:24; C~11 ˆ 1; C~12 ˆ 100;

m1 ˆ 0:40 and m2 ˆ 0:24

…15a†

…15b†

for Eq. (9). Then Eq. (12) may be evaluated and plotted in part (b) of Fig. 1 as a function of s, for 1 P s P 0. Obviously, no extra zero of d(s) can be found. Thus, Eqs. (13) and (14) themselves turn out to be the best solutions. They are plotted in part (a) of Figs. 2 and 3, respectively. One can see that Eqs. (13) and (14) reproduce the Monte Carlo simulation results very precisely. 2.1.2. Dominance Parameters in Eq. (8) for Monte Carlo simulations in Ref. [9] are m11 ˆ 0:30;

m12 ˆ 0:40; m21 ˆ 0:21; C~11 ˆ 1; C~12 ˆ 100;

m22 ˆ 0:24; C~21 ˆ 1; C~22 ˆ 100:

Since C~12 ˆ C~22  C~11 ˆ C~21 , the most frequent collision is that atom-1 strikes atom-2 and atom-2 strikes atom-2. The species 2 behaves as in a monatomic medium, while the energy distribution of species 1 is determined by that of species 2 in a complicated way. This case is called the ``domi-

256

Z.L. Zhang / Nucl. Instr. and Meth. in Phys. Res. B 152 (1999) 252±266

Fig. 1. Calculated results of d(s): de®ned by Eq. (12), as a function of s. (a) Curve A stands for detailed balance, Curve B stands for ignorance and detailed balance; (b) Dominance; (c) Ignorance.

nance'' of species 2 over species 1 [9]. Therefore, it is natural to choose parameters (15b) for Eq. (9) then Eq. (12) can be evaluated and plotted in part (b) of Fig. 1 as a function of s, for 1 P s P 0. Two extra zeros of d(s) can be found, s0 ˆ 0.3986 and s1 ˆ 0. Taking account of contributions of s ˆ 1, s0 and s1 , Eqs. (10) and (11) turn out to be w11 …x† ˆ w1 …x†…1 ÿ 0:6068xa ‡ 140:371x†;

w12 …x† ˆ w2 …x†…1 ÿ 0:5869xa ÿ 0:6649x†; w22 …x† ˆ w2 …x†…1 ‡ 6:5548
10ÿ3 xa

…16†

f11 …x† ˆ f1 …x†…1 ÿ 0:2906xa †; f12 …x† ˆ 100f11 …x†; f21 …x† ˆ f1 …x†…1 ‡ 3:2552
10ÿ3 xa †; f22 …x† ˆ 100f21 …x†;

uted by the third highest pole s1 ˆ 0, i.e. the third term in the bracket of the ®rst relation of Eq. (16). This term cannot be predicted by the UC theory [9]. 2.1.3. Ignorance Parameters in Eq. (8) for Monte Carlo Simulations in Ref. [9] are

w21 …x† ˆ w1 …x†…1 ‡ 6:7748
10ÿ3 xa ÿ 1:4037x†; ‡ 6:6472
10ÿ3 x†;

Fig. 2. Normalized energy distributions xwij …x† vs. reduced energy x ˆ E0 /E in a collision cascade in a binary medium using Lindhard collision cross sections: Thick: Monte Carlo Simulation results are copied from [9]; Thin: SigmundÕs analytical theory Eq. (10); (a) Detailed balance, calculated by Eq. (13); (b) Dominance, calculated by Eq. (16); (c) Ignorance, calculated by Eq. (18); (d) Ignorance and detailed balance, calculated by Eq. (13).

…17†

where a ˆ 1 ÿ s0 ˆ 0:6014. Eqs. (16) and (17) are calculated and plotted in part (b) of Figs. 2 and 3, respectively. One can see that these solutions reproduce the Monte Carlo simulation results very well. Particularly, for xw11 …x†, the large deviation from the asymptotic solution xw1 …x† is contrib-

m11 ˆ 0:40

m12 ˆ 0:30 m21 ˆ 0:20 C~11 ˆ 1 C~12 ˆ 0:01

m22 ˆ 0:24 C~21 ˆ 0:01 C~22 ˆ 1:

Since C~11 ˆ C~22  C~12 ˆ C~21 , the most frequent collision is that atom-1 strikes atom-1 and atom-2 strikes atom-2, each species shows the same energy dependence as in a monatomic medium. Therefore, this case is called ``ignorance'', it is natural to choose Eq. (15b) for Eq. (9), then, Eq. (12) can be evaluated and plotted in part (c) of Fig. 1 as a function of s, for 1 P s P 0. One extra zero of d(s) has been found, s0 ˆ 0.9909415. Taking account of

Z.L. Zhang / Nucl. Instr. and Meth. in Phys. Res. B 152 (1999) 252±266

257

ther may be neglected in the energy window concerned, which is the reason for the failure of the UC theory [9] in this regime. 2.1.4. Ignorance and detailed balance The parameters in Eq. (8) for the Monte Carlo simulations in Ref. [9] are m11 ˆ 0:40;

m12 ˆ 0:30;

m21 ˆ 0:20;

m22 ˆ 0:24 ± ignorance; C~11 ˆ 1; C~12 ˆ 100; C~21 ˆ 100;

Fig. 3. Normalized recoil densities x2 fij (x) vs. reduced energy x ˆ E0 /E in a collision cascade a in a binary medium using Lindhard collision cross sections: Thick: Monte Carlo Simulation results are copied from [9]; Thin: SigmundÔs analytical theory Eq. (11); (a) Detailed balance, calculated by Eq. (14); (b) Dominance, calculated by Eq. (17); (c) Ignorance, calculated by Eq. (19).

C~22 ˆ 1

± detailed balance:

Due to the same reason as in detailed balance, it is natural to choose m1 ˆ 0.30 and m2 ˆ 0.2 for Eq. (9). Thus, Eq. (12) is plotted in part (a) of Fig. 1. No extra zero of d(s) can be found. Therefore, Eq. (13) itself turns out to be the best solution. Eq. (13) is plotted in part (d) of Fig. 2. One can see that Eq. (13) reproduces the Monte Carlo simulation results very well. On the contrary, it is not possible for the UC theory to predict the simulation results in the relevant energy windows [9]. 2.2. Comparison with the VCU theory [11]

contributions of both poles s ˆ 1 and s0 , Eqs. (10) and (11) turn out to be, w11 …x† ˆ w1 …x†…1 ‡ 0:8306xa †; w12 …x† ˆ w2 …x†…1 ÿ 0:99554xa †; w21 …x† ˆ w1 …x†…1 ÿ 0:99234xa †;

…18†

w22 …x† ˆ w2 …x†…1 ‡ 1:1893xa †; f11 …x† ˆ f1 …x†…1 ‡ 0:8249xa †; f12 …x† ˆ f2 …x†…1 ÿ 0:9892xa †; a

f21 …x† ˆ f1 …x†…1 ÿ 0:9856x †; f22 …x† ˆ f2 …x†…1 ‡ 1:1817xa †;

In 1993, Vicanek et al. proposed a general method to study energy distributions of recoil atoms in collision cascades in composite media [11]. According to the VCU theory, linear transport equations describing the particle ¯ux are reduced to a computationally much simpler system of differential equations, X  ÿ ak Sjk …E0 †Wij …E0 † ‡ aj Skj …E0 †Wik …E0 † N k

…19†

where a  1 ÿ s0 ˆ 0:0090585, Eqs. (18) and (19) are calculated and plotted in part (c) of Figs. 2 and 3, respectively. One can see that w11 ; w22 ; w21 ; f11 and f22 reproduce the Monte Carlo simulation results accurately, and w12 ; f12 and f21 agree reasonably well with the simulation results. Because the ®rst and second highest poles (s0 1) provide almost identical terms in magnitude, nei-

Xh d N ak rsjk …E0 †E02 Wij …E0 † dE0 k i r ‡aj rkj …E0 †E02 Wik …E0 † ‡ E0 dij d…E0 ÿ E† ˆ 0 ‡

…20† with the obvious initial condition, Wij …E0 > E† ˆ 0;

…21†

where Wij …~ t0 † ˆ t0 Gij …~ t;~ t0 †. The stopping crosssection Sij (E) and the energy slowing down cross rsij …E† and rrij …E† have been introduced by Urbas-

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Z.L. Zhang / Nucl. Instr. and Meth. in Phys. Res. B 152 (1999) 252±266

sek et al. in Ref. [11]. In this section, let us use the VCU theory to restudy the binary system (a1 ‡ a2 ˆ 1) with the power cross-section Eq. (9) Thus, Eq. (20) can be solved exactly, ( wi1 … x† ˆ w1 … x†…1 ‡ Ci S1 xa † i ˆ 1; 2; …22† wi2 … x† ˆ w2 … x†…1 ÿ Ci S2 xa † where Si , ri and a are constants de®ned by Si ˆ C~ii Cmi =…C~i1 ‡ C~ir †;

ri ˆ Si =…1 ÿ mi †

a ˆ …S1 ‡ S2 †=…1 ÿ r1 ÿ r2 †:

3.1. Di€erential particle ¯ux The procedure of determining the asymptotic solutions (E  E0 ) is a generalization of the one described by Andersen and Sigmund in Ref. [7]. The main problem is to ®nd the highest single pole of Eqs. (5) and (6). The poles may occur at the zeros of the determinant DL …s†. Using Eq. (7), we can easily prove that the highest zero of DL …s† is s ˆ (1ÿ0.5L) for L ˆ 0,1. Therefore, the asymptotic solutions are given by

…23†

Ci is an integral constant. Obviously, Ci cannot be determined by the initial condition (21) alone. Actually, the authors of the VCU theory have never given any pertinent method to evaluate these integral constants. On the one hand, inserting Ci ˆ 0, Eq. (22) is reduced to Eq. (13), which cannot predict the simulation results in the cases of dominance and ignorance. On the other hand, if Ci 6ˆ 0; a de®ned by Eq. (23) could be a negative number, it could bring about a catastrophic result, because, the term xa in Eq. (22) will provide a divergent correction. Detailed balance is a typical example. Previously, it was show that the asymptotic solutions (13) and (14) may reproduce the corresponding simulation results accurately. Unfortunately, inserting Eq. (15a) into Eq. (23) yields a ˆ ÿ2:720, which is impossible. Furthermore, even if the scattering cross-section (8) is used instead of Eq. (9) in the analysis, Eq. (20) can be solved numerically. It is found that the abovementioned conclusions remain unchanged. Therefore, at least for some cases, the VCU theory cannot yield reasonable correction terms in Eq. (22), no matter what integral constant Ci is chosen. 3. Statistical distribution functions The excellent agreement between SigmundÕs theory and Monte Carlo simulation results [9] stimulates me to go further into this area for searching anisotropic statistical distribution functions, even though the pioneering work has been done by Sigmund et al. [10,12].

Gij …E; E0 † ˆ

1 Dj E
 Gj …E0 †; N t0 E0 Qj …E0 † D E0

…24†

Fij …E; E0 † ˆ

E Dj X

b  Fj …E0 † E02 D k jk

…25†

for L ˆ 0, and G1ij …E; E0 † ˆ

D1j Pi 1
1
; N t0 E0 Qj …E0 † D P0j

…26†

Fij1 …E; E0 † ˆ

Pi D1j X
…Ujk bjk † E0 P0j D1 k

…27†

for L ˆ 1. Here X X L DL ˆ DLk bkj eLkj ; DLj



j

k

 0:5L ˆ D …1 ÿ 0:5L† ji …Mj =Mi † ; L

bLij  bLij …1 ÿ 0:5L†; dhl i eLij  e …s† sˆ…1ÿ0:5L† ds ij p ~ n 2Mi E; Pi ˆ ~

~ P0j ˆ ~ n0

p 2Mj E0 :

Eqs. (24) and (25) were given in Ref. [13]. Eqs. (24) and (26) allows to rederive or con®rm some basic results for particle ¯uxes in Ref. [12], X ‰aj Skj …E0 †Gk …E0 † ÿ ak Sjk …E0 †Gj …E0 †Š ˆ 0 …28† k

X k

‰Ukj aj Skj …E0 †G1jk …E; E0 † ÿ …Mj =Mk †0:5 Ujk ak Sjk …E0 †G1jj …E; E0 †Š ˆ 0

…29†

Z.L. Zhang / Nucl. Instr. and Meth. in Phys. Res. B 152 (1999) 252±266

Inserting Eqs. (24) and (26) into Eq. (3), only the L ˆ 0,1 terms are considered. We obtain " # ~ E Pi n0
~ t;~ t0 † ˆ Kj
‡ 3Aj ; …30† Gij …~ E0 P0j where 1 Dj
; Kj  N t0 E0 Qj …E0 † D

  t00 ÿ~ t0 ;  h…E1  ÿT †d ~

…31†

1

ÿ

b112 e112

for reproduction of the cascade. Thus, these functions are related to particle ¯uxes in the following way   Fij ~ t;~ t0  Z   XZ 3 tp Wik ~ tp tp ;~ d~ t0 ;~ t00 drkj ~ ˆ N aj k

D D1j Aj ˆ 1
: D Dj

For a binary medium Eq. (30) reads, " # ~ E Pi n0
~ t;~ t0 † ˆ K 1 ‡ 3 p
A Gi1 …~ i ˆ 1; 2; E0 2M2 E0 " # ~ E Pi n0
~ t;~ t0 † ˆ K 2 ‡ 3 p
A ; i ˆ 1; 2; Gi2 …~ E0 2M1 E0

where

259




D ˆ U21 b21 b11 e11 ‡ ÿ
‡ U12 b12 b121 e121 ‡ b22 e22 ; 1 D A  p
1 : c12 D Eq. (31) was ®rst derived by Sigmund et al. for a binary medium with nearly equal masses (M1 M2 ) [10]. Furthermore, replacing E and ~ Pi by deposited energy and momentum spatial distribution functions respectively Eq. (31) turns out to be the form which was given by Sigmund et al. recently [12]. 3.2. Freezing recoil, scattering and collision densities Freezing recoil, scattering and collision densities were introduced and derived asymptotically by Sanders et al. for a monatomic medium [3,4]. Following their track, let us take into account only those recoils, which are either excited into ÿ
~ t0 from zero energy or which lose so much t0 ; d3~ energy in a single collision with a particle at rest that from an energy >E1 they ÿ slow
down into the t0 . In this way, speci®ed velocity interval ~ t0 ; d3~ the energy evolution of the cascade is stopped at energy E1 and a ``frozen in'' picture of the cascade for E0 6 E1 < E is obtained. E1 acts as a threshold

…32†

  t;~ t0 Fij ~  Z   X Z 3 drjk ~ tp Wij ~ tp tp ;~ ak d ~ t0 ;~ t00 ˆN k

  t0 ÿ~ t0 ;  h…E1  ÿE ‡ T †d ~

…33†

      t;~ t0 ˆ Fij ~ t;~ t0 ‡ Fij ~ t;~ t0 : Fijc ~

…34†

For cross-sections like Eq. (4), following the previous procedure, Eqs. (32)±(34) may derived asymptotically for E  E1 P E0 , E 1X Fij …E; E0 † ˆ
Dk Rkj  Fj …E0 †; …35† E1 E 0 D k Fij …E; E0 † ˆ

E Dj X
Wjk  Fj …E0 † E1 E 0 D k

…36†

for L ˆ 0 and Fij1 …E; E0 † ˆ

Fij1 …E; E0 †

Pi 1 X 1
1 D Ukj Rkj ; E1 P0j D k k

…37†

r E1 E0

…38†

Pi D1j X 1 ˆ
W E1 P0j D1 k jk

for L ˆ 1. Here   Z E0 =cE1 cE1 dX Aij …X †X ; Rij ˆ ai
E0 0  Wij ˆ aj
 Wij1 ˆ aj

E1 E0

E1 E0

Z

1 1…1ÿE =E† 0 c

0:5 Z

dX Aij …X †…1 ÿ cX †;

1 1…1ÿE =E† 0 c

dX Aij …X †…1 ÿ X =Uji †:

260

Z.L. Zhang / Nucl. Instr. and Meth. in Phys. Res. B 152 (1999) 252±266

Obviously, if E1 ˆ E0 , Eqs. (35) and (36) reduce to the forms which were given in [13], Eqs. (35) and (37) can be simpli®ed to Eqs. (25) and (27) respectively.

Nij …~ t; E1 † Z Z E X dE0 N aj Wik …E0 † ˆ

3.3. Energy and momentum sharing, recoil and scattering numbers

Nij …~ t; E1 † Z E X Z 0 0 ˆ dE Wij …E †N ak

According to the frozen in picture, it is natural to de®ne energy and momentum sharing [10±12], recoil and scattering numbers, respectively, Z E1 t0
E0
Fijc …~ t; E1 † ˆ d3~ t;~ t0 †; …39† F…ij† …~ 0

~ t; E1 † ˆ P…ij† …~

Z 0

Z t; E1 † ˆ Nij …~

E1

E1

0

Z

Nij …~ t; E1 † ˆ

E1

0

t0
Mj~ t0
Fijc …~ d3~ t;~ t0 †;

…40†

t0
Fij …~ d3~ t;~ t0 †;

…41†

t0
Fij …~ d3~ t;~ t0 †:

…42†

It is straightforward to show that Eqs. (39) and (40) satisfy the laws of conservation of energy and momentum, X X ~ F…ij† …~ t; E1 † ˆ E; t; E1 † ˆ ~ Pi P…ij† …~ j

j

E1

k

k

E E1

dE0 ‰Wik …E0 †aj

‡ Wij …E0 †ak

Z

E0 E0 ÿE

Z 0

E1

drjk …E0 ; T †…E0 ÿ T †Š;

~ t; E1 † P…ij† …~ ˆ~ n2N

XZ k

‡

E E1

dE0 ‰Ukj G1ik …E; E0 †aj

G1ij …E; E0 †ak

Z

E0

E0 ÿE

Z 0

E1

E E0 ÿE1

drjk …E0 ; T †;

…45†

…46†

 F…j† …E1 †

…47†

~ Pi X h ~ P…ij† …~ t; E1 † ˆ 1
ÿ D1k Ujk Bjk D k i 1 ‡Dj …b1jk e1jk ‡ Ujk Bjk †

…48†

t; E1 † ˆ Nij …~

Nij …~ t; E1 † ˆ drkj …E0 ; T †T

drkj …E0 ; T †;

respectively. Eqs. (43)±(46) are exact formulae for arbitrary cross-sections, (43), (45) and (46) are given by Urbassek et al. in Ref. [11] without proof. However, replacing the total ¯ux by the spatially dependent particle ¯ux, Eqs. (43) and (44) turn out to be the forms which were given by Sigmund et al. recently Ref. [12]. For cross-sections (4), following the previous procedure, these functions can be derived asymptotically for E  E1 ,  E X F…ij† …~ t; E1 † ˆ
ÿ Dk Bkj ‡ Dj …bjk ejk ‡Bjk † D k

Inserting Eqs. (32)±(34) into Eqs. (39)±(42), we obtain t; E1 † F…ij† …~ XZ ˆN

0

k

E1

E1

 E 1X 
Dk bkj …0† ÿ bkj E1 D k  Nj …E1 †

…49†

E Dj X
b  Nj …E1 † E1 D k jk

…50†

where …43†

BLij 

d L b …s†j ds ij sˆ…1ÿ0:5L†

for L ˆ 0; 1:

Inserting Eq. (24) into Eq. (47) and Eq. (50) yields X F…j† …E1 † ˆ N ‰ak rsjk …E1 †E12 Wj …E1 † drkj …E0 ; T †T

drjk …E0 ; T †…E0 ÿ Ujk T †Š; …44†

k

‡ aj rrkj …E1 †E12 Wk …E1 †Š;  …E1 † ˆ Wj …E1 †N N…j†

X ‰ak Sjk …E1 †Š: k

…51† …52†

Z.L. Zhang / Nucl. Instr. and Meth. in Phys. Res. B 152 (1999) 252±266

E a2 S12 …E0 † NE02 T …E0 †

3.4. Approximate asymptotic solutions for arbitrary cross-sections [11]

W2 …E0 † ˆ

It was demonstrated that SigmundÕs linear transport equations can be solved for cross-sections like Eq. (4). Here the SC theory is used to build up approximate asymptotic solutions for arbitrary cross-sections drij … E; T †. The SC theory consists of two steps. The ®rst step is the ``separating approximation'', i.e. using cross-sections Eq. (4) to approximate drij … E; T † in these equations. Thus, we obtain all the asymptotic solutions. The second step is ``combining approximation'', i.e. replacing Qi …†Aij … X †d X by drij …; T † in asymptotic solutions gained in the ®rst step. Specifically, the following replacements () are necessary for building up SC solutions:

E a1 S21 …E0 † ‰a1 S11 …E0 † ‡ a2 S12 …E0 †Š; E03 T …E0 † E a2 S12 …E0 † ‰a1 S21 …E0 † ‡ a2 S22 …E0 †Š; F2 …E0 † ˆ 3 E0 T …E0 †

Qi …†bij  aj Sij …†;

aj r~ij …†;

etc:

j

Using the SC theory, Eqs. (28) and (29) and Eqs. (51) and (52) turn out to be approximate asymptotic solutions for arbitrary cross-sections. For a binary target, Eqs. (28) and (29) read a2 S12 …E0 †W1 …E0 † ˆ a1 S21 …E0 †W2 …E0 †;

…53†

…56† where r1 …E0 † ‡ a2 S12 …E0 †~ r2 …E0 †: T …E0 † ˆ a1 S21 …E0 †~ Eq. (55) is also given in Ref. [11]. Eqs. (55) and (56) have been shown to ®t by simulation solutions [9] reasonably well in the asymptotic regime of small recoil energies for general power cross-sections Eq. (8) except in the case of ignorance.

0:5

ˆ …M2 =M1 † U21 a1 S21 …E0 †G1j2 …E; E0 †

…54†

respectively (j ˆ 1,2). Eqs. (51)±(53) were given by Urbassek et al. in Ref. [11] without proof. If the E1 dependence of Fij and ~ P…ij† are insigni®cant, Eqs. (28) and (29) can be rederived [12]. In addition, applying the SC theory to Eqs. (24) and (25), we obtain, E a1 S21 …E0 † ; NE02 T …E0 †

For a power potential (Kij rÿ1=m ) interaction, there are three major scattering cross-sections [13], each of them being a special case of Eq. (4). Thus, all the asymptotic solutions in Section 3 may be written down easily for the present case. (i) Present theory for 1 P m > 0 Qi …E† ˆ Eÿ2m ; Aij …X † ˆ Cij cÿm Zm …X †X ÿ1ÿm ;  2m  m cs Kij Mi
: Cij ˆ pm m Mj

…57†

(ii) Present theory for m ˆ 0

U12 a2 S12 …E0 †G1j1 …E; E0 †

W1 …E0 † ˆ

F1 …E0 † ˆ

4. Asymptotic solutions for power potential interaction

Qi …†eij bij  aj ‰rsij …† ‡ rrij …†Š  aj r~ij …† X

…55†

Qi …†Bij  ÿaj rrij …†

d Qi …† ‰bij …s†eij …s†Šjsˆ1  aj rsij …† ds

r~i …† ˆ

261

Qi …E† ˆ 1

and

Aij …X † ˆ rij

then, we have b1ij …s† ˆ

aj rij cs ; …s ‡ 1:5† 

e1ij …s†

h i s‡1:5 …e ‡ 1:5†c ÿ Uij 1 ÿ …1ÿc†  h i s‡0:5 …s ‡ 1:5† ÿVij 1 ÿ …1 ÿ c† ; …s ‡ 0:5†

ˆc

ÿsÿ1

…58†

262

e1ij

Z.L. Zhang / Nucl. Instr. and Meth. in Phys. Res. B 152 (1999) 252±266

# r " 2 Mj 1 ÿ1 ln…1 ÿ c† ÿ ln c
Mi c     Vij 1 ‡ p 2 ÿ 1 ln…1 ÿ c† ‡ 1 : c c

1 ˆ p ‡ c

(iii) Sigmund theory for 1 P m P 0 Qi …E† ˆ Eÿ2m Qij …X † ˆ Cij c

and ÿm

X ÿ1ÿm

…59†

Thus, we have b1ij …s† ˆ aj Cij e1ij …s†

csÿm ; …s ‡ 0:5 ÿ m†

r Mj eij …s ‡ 0:5† ˆ Mi Vij …s ‡ 0:5 ÿ m† ‡ p eij …s ÿ 0:5† : …s ÿ 0:5 ÿ m† c

bij and eij etc. have been given in Refs. [7,13] for L ˆ 0. 4.1. Generalized Sigmund formula [7,13] If the power potential satis®es m1 ˆ m2 ˆ


ˆ mn ˆ m; Kij ˆ Kji ; rij ˆ rji

…60†

then, by inserting Eqs. (57)±(59) into Eq. (6a), we obtain   bLij …s† aj Mi 2m ˆ ; 1 P m P 0: …61† bLji …s† ai Mj On the other hand, using Eq. (61), we derived generalized SigmundÕs relations [7,13],   DLj aj Mi 2mÿL ˆ for L ˆ 0; 1: …62† DLi ai M j Inserting Eq. (62) into Eq. (30) yields r r  E E Mj t;~ t0 † ˆ Kj ‡ 3…~ n
~ n0 †
Ai Gij …~ Mi E0 E0

…63†

However, Eq. (63) is valid for a multispecies medium in both the Sigmund and Present theories.

4.2. Range of validity of the asymptotic solutions [7] The validity of the asymptotic solutions in Section. 3 is directly related to the correction terms in the asymptotic expansion of solutions. These correction terms are determined by the residues at subsequent poles of GLij …s†, i.e. zeros of DL (s), in most cases. For a binary medium, we have   DL …s† ˆ bL11 …s† eL11 …s† ÿ 1 bL21 …s†eL21 …s†   ‡ bL22 …s† eL22 …s† ÿ 1 bL12 …s†eL12 …s†     ‡ bL11 …s† eL11 …s† ÿ 1 bL22 …s† eL22 …s† ÿ 1   ‡ bL12 …s†bL21 …s† eL21 …s†eL21 …s† ÿ 1 : For simplicity, only the case of m ˆ 0 is treated. Thus,   drij …E; T † z ˆ rdT =Tm ;   drij …E; T † p ˆ C0 dT =T where [H]z stands for H in the Present theory, and [H]p stands for H in the Sigmund theory. The extra zero s0 of DL (s) may be found easily. For L ˆ 0, s0 is calculated and plotted in Fig. 4 as a function of the concentration a1 …or a2 † for three typical binary systems HfN, ZrN and TiN in both the Sigmund and Present theories. One can see ‰s0 Šp  0:5

but

‰s0 ŠE < 0

for TiN and ZrN This fact shows that the asymptotic solutions (L ˆ 0) may not be suitable for HfN and ZrN in the Sigmund theory, but are acceptable in the Present theory. Therefore, the Present theory has a much larger range of validity. For further simplicity, let us only consider a1 ˆ a2 ˆ 0:5. In the Sigmund theory, Eq. (5) reduce to   s 1 1 ~ …s† ˆ  ; …64† G N t0 C0 E0 Dÿ …s† D‡ …s† where G‡ ˆ G11 ˆ G22 ;

Gÿ ˆ G12 ˆ G21 ;

D …s† ˆ e11 …s† ÿ 1 ‡ c2 ‰e12 …s†  1Š;

c ˆ c12 :

Z.L. Zhang / Nucl. Instr. and Meth. in Phys. Res. B 152 (1999) 252±266

263

G p2 1 s0 …s11 ‡ s12 †; ; Aˆ 0 ˆ 6 e11 ‡ ce12 D‡ …s0 † …G† d D …s† : D0 …a† ˆ ds sˆa Therefore, the relative discrepancy between Eqs. (65) and (24) is given by D ˆ Ax1ÿs0

…66†

which is calculated and plotted in Fig. 5 as a function of x ˆ E0 /E for the HfN and HfC systems. In the Present theory, Eq. (5) is reduced to   1 1 ~ ˆ c…s ‡ 1†
 …67† G N t0 rE0 Dÿ …s† D‡ …s† for L ˆ 0, where D …s† ˆ 2cs ÿ 1‡ …1 ÿ c†s‡1  cs‡1 . A simple calculation shows that Dÿ (s) has one positive zero s ˆ 1 and D‡ (s) has at least one zero s ˆ 0. Due to the complication of the second term of Eq. (67), let us consider the following three cases, p Case I : c ˆ 0:5 i.e. M1 =M2 ˆ 3  2 2, Eq. (67) reads

Fig. 4. The additional positive zero of D(s), as a function of the chemical concentration a…N † for three typical systems TiN, ZrN and HfN. In Present theory, the highest additional zero is s0 ˆ 0.174 for HfN. No such zero exit for ZrN and TiN. In Sigmund theory, the highest additional zero is s0 0.5 for ZrN and HfN [7].

A simple calculation shows that Dÿ (s) has one positive zero s ˆ 1 and D‡ (s) has one positive zero s0 , s0 is evaluated as a function of c. The calculated results show that, s0  0, only for c  1 or M1  M2 , thus the asymptotic solutions Eqs. (24) and (26) are acceptable [10]. Taking account of the contribution of s0 , the inverse Laplace transforms of Eq. (64) are given by G …E0 † ˆ where

C0 E G
…1  Ax1ÿs0 †;
N t0 C0 E02 …G†

…65†

Fig. 5. The relative errors D are de®ned by Eq. (66) in Sigmund theory, and by Eq. (71) in Present theory, as functions of x ˆ E0 /E. Obviously, the Present theory can give a more accurate asymptotic solution Eq. (24).

264

Z.L. Zhang / Nucl. Instr. and Meth. in Phys. Res. B 152 (1999) 252±266

  1 1 ~ ˆ …s ‡ 1† :  G 2N t0 rE0 s ÿ 1 s ÿ 1 ‡ 2ÿs Thus, D‡ (s)/(s+1) has only one real zero s ˆ 0, therefore, E …1  Ax†; 2N t0 rE02 1 ˆ 1:629 . . . Aˆ 2…1 ÿ ln 2† G …E0 † ˆ

…68†

Case II : c ˆ c0 , while c0 satis®es

accurate asymptotic solution Eq. (24) than the one given by the Sigmund theory for the same system. For instance, E0 /E ˆ 0.01, ‰DŠE  10%, but ‰DŠp  30% . Therefore, in the Sigmund theory, the correction term in the asymptotic expansions of solution for L ˆ 0 not only could be as large as the leading term for L ˆ 1, but could also disturb the isotropic leading term for L ˆ 0, thus complicating the sputtering yield energy distribution in the case where c is very small [17].

2c ‡ …1 ÿ c† ln …1 ÿ c† ‡ c ln c ˆ 0

4.3. Exact solutions

or

Another remarkable feature of hard-sphere scattering cross-section is that all the transport equations can be solved analytically, even exactly. For a monatomic target bombarded by a self-atom, Eqs. (5) and (6) is reduced to

c0 ˆ 0:309; . . . ; i:e: …M1 =M2 †

1

ˆ 10:84 . . .

Thus, D‡ (s)s2 for s0. Therefore, G …E0 † ˆ

2E G …1  Ax ln x†; N t0 rE02 …G†

…69†

where G=…G† ˆ 0:558; . . . ; A ˆ 1:06 . . . Case III: c 6ˆ 0:5; c0 : A simple calculation shows that D‡ (s) has a nonvanishing zero, s0 < 0 for 1 P c > c0 and s0 > 0 for c0 P c > 0. Taking account of the contributions of these three poles s ˆ 0,1 and s0 , the inverse Laplace transforms of Eq. (67) are given by G …E0 † ˆ

2E G …1  Ax1ÿs0  Bx†; N t0 rE02 …G†

…70†

where G c ; ˆ 0 …G† Dÿ …1† …s0 ‡ 1† D0ÿ …1† Aˆ 2 D0‡ …s0 †

  L0 1 L L ~ ~ ‡ F …s† ; G …s† ˆ t0 2E0 F~L …s† ˆ F~L …s†

R1

p dxx PL … x† 1 0 i: ˆ h E0 1 ÿ 2 R 1 dxxs PL …px† 0

…72†

s

Taking account of the contributions of all poles, including complex variable poles of Eq. (72), the inverse Laplace transforms of Eq. (72) are given exactly as follows. f …x† ˆ 1; f 1 …x† ˆ x0:5 ;

and

1 D0ÿ …1† Bˆÿ : 2 D0‡ …0†

Writing Eqs. (68)±(70) together, the relative discrepancy is given by 8 Case I > < Ax D ˆ ÿAx ln x Case II …71† > : 1ÿs0 Ax ÿ Bx Case III which is calculated and plotted in Fig. 5 as a function of x for the HfN and HfC systems. Fig. 5 shows that the Present theory generates a more

2

1:5

f …x† ˆ x cos

! p 3 ln x ; 2

! r p 15 2 11 ln x ‡ U0 ; f …x† ˆ x sin 2 11 p 11 ÿ1 etc: U0 ˆ tan 2 3

Substituting F L …E; E0 † ˆ …E=E02 †f L …E0 =E† into Eq. (3), the numerical calculation has shown that Eq. (3) converges very well as L increases in®nitely.

Z.L. Zhang / Nucl. Instr. and Meth. in Phys. Res. B 152 (1999) 252±266

265

5. Summary

Acknowledgements

(i) It is demonstrated that the theory of Sigmund et al. (Eqs. (10) and (11)) agrees very well with the Monte Carlo simulation results [9]. This means that the Monte Carlo simulations [9] provide a strong support for SigmundÕs theory [7] and SigmundÕs theory give these simulations a perfectly rational explanation. It is shown that both the UC theory [9] and the VCU theory [11] are incorrect, because these theories cannot reproduce some of the simulations. (ii) It is demonstrated that all the transport equations describing the statistics of the linear random collision cascades are suitable for a complete use of both simple power and exact scattering cross-sections [13]. Asymptotic formulae for the anisotropic particle ¯ux (L ˆ 0,1) have been derived for any multicomponent medium by a nonperturbation theory [10,12]. In addition, asymptotic formulae for energy and momentum sharing are presented. (iii) The SC approximation is presented to build up approximate asymptotic solutions for arbitrary cross-sections Ref. [11], based on asymptotic solutions in Section 3. Some of the analytical results given in Refs. [11] and [12] are con®rmed. (iv) By the use of the cross-sections dr ˆ C0 dT =T and dr ˆ rdT =Tm , the correction terms in asymptotic expansions of the isotropic solution (L ˆ 0) have been evaluated for a binary target. The additional positive pole of the Laplace transform of the isotropic ¯ux is s0  0:5 for most cases in the Sigmund theory [7], or even higher for some cases. But the pole is much lower in the Present theory, even for a binary target with M1 / M2 >10 such as HfN, where s0 ˆ 0.17. Therefore, the Present theory may give much more accurate asymptotic solutions. In other words, the exact cross-section enlarges the range of validity of the asymptotic solution as compared to the simple power law cross-section [7]. (v) For a monatomic target bombarded by a self-atom the transport equations can be solved exactly, by the use of hard sphere collision [13].

I would like to thank Dr. K.R. Padmanabhan, Associate Professor, Department of Physics and Astronomy of Wayne State University, USA, without whose initial supervision (1983±1986), this work could not have been done. Special thanks are due to Dr. P.K. Kuo, Professor, Department of Physics and Astronomy of Wayne State University, USA, for recommending me as a graduate student in the Department in 1980. I wish to thank my wife Bin Chu Zhang, Engineer in the Department of Mechanical Engineering of Huainan Institute of Technology, PRC, for her constant encouragement and support. I like to thank my son Lai Zhang, for giving me some computer assistance. Special thanks are due to Professor Zhen Hai Feng, ex-president of Huainan Institute of Technology for providing the necessary conditions to complete this work. I am very grateful to the referee for many critical comments on the manuscript. Also, Professor H.H. Andersen, editor of NIM-B, deserves special thanks for his constant encouragement during the revision of the paper and for the attention and criticism paid in reading the manuscript. References [1] P. Sigmund, Phys. Rev. 184 (1969) 383. [2] P. Sigmund, Phys. Rev. 187 (1969) 768. [3] W. Huang, H.M. Urbassek, P. Sigmund, Philos. Mag. A 52 (6) (1985) 735. [4] J.B. Sanders, H.E. Roosendaal, Radiat. E€. 24 (1975) 161. [5] H.E. Roosendaal, U. Littmark, J.B. Sanders, Phys. Rev. B 26 (9) (1982) 5261. [6] P. Sigmund, in: R. Behrisch (Ed.), Sputtering by Particle Bombardment I, Topic in Applied Physics, vol. 47, Springer, Berlin, Heidelberg, New York, 1981, p. 65. [7] N. Andersen, P. Sigmund, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 39 (3) (1974). [8] P. Sigmund, Nucl. Instr. and Meth. B 18 (1987) 375. [9] H.M. Urbassek, U. Conrad, Nucl. Instr. and Meth. B 69 (1992) 413. [10] P. Sigmund, M.W. Sckerl, Nucl. Instr. and Meth. B 82 (1993) 242. [11] M. Vicanek, U. Conrad, H.M. Urbassek, Phys. Rev. B 47 (2) (1993) 617.

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Z.L. Zhang / Nucl. Instr. and Meth. in Phys. Res. B 152 (1999) 252±266

[12] M.W. Sckerl, P. Sigmund, M. Vicanek, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 44 (3) (1996) 1. [13] Z.L. Zhang, Nucl. Instr. and Meth. B 149 (1999) 272. [14] F. Keywell, Phys. Rev. 87 (1952) 160.

[15] F. Keywell, Phys. Rev. 97 (1955) 1611. [16] W. Brandt, R. Laubert, Nucl. Instr. and Meth. 47 (1976) 201. [17] M. Szymonski, Phys. Lett. 82A (1981) 203.